Stevenson 18 Management of Waiting Lines 18 2

Stevenson 18 Management of Waiting Lines

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Learning Objectives § § § § § Explain why waiting lines form in systems. Implications of waiting line Goal of waiting line management Characteristics of waiting line Measures of waiting line performance Queuing models – Infinite Source and Finite Source Constraint management Psychology of waiting Operations strategy 18 -3

Why do waiting lines form? § Waiting lines occur when there is a temporary imbalance between supply (capacity) and demand. § Waiting lines add to the cost of operation and they reflect negatively on customer service, § It is important to balance the cost of having customers wait with the cost of providing service capacity. § From a managerial perspective, the key is to determine the balance that will provide an adequate level of service at a reasonable cost. 18 -4

Disney World § Waiting in lines does not add enjoyment § Waiting in lines does not generate revenue Waiting lines are non-value added occurrences 18 -5

Waiting lines § Waiting Time: Operators and machines waiting for parts or work to arrive from suppliers or other operations. Customers waiting in line. § One of the “seven wastes” 18 -6

Waiting Lines § Queuing theory: Mathematical approach to the analysis of waiting lines. § Goal of queuing analysis is to minimize the sum of two costs § Customer waiting costs § Service capacity costs 18 -7

Goal of waiting line management Minimize the sum of two costs: customer waiting costs and service capacity costs Cost Total cost = Customer waiting cost + Total cost Capacity cost Cost of service capacity Cost of customers waiting Service capacity Optimum 18 -8

Implications of Waiting Lines § Cost to provide waiting space § Loss of business § Customers leaving § Customers refusing to wait § Loss of goodwill § Reduction in customer satisfaction § Congestion may disrupt other business operations 18 -9

System Characteristics § Population Source § Infinite source: customer arrivals are unrestricted § Finite source: number of potential customers is limited § Number of Servers (channels) § Arrival and service patterns § Queue discipline (order of service) 18 -10

System Characteristics: Number of servers/phases Multiple channel Channel: A server in a service system Multiple phase 18 -11

System Characteristics: Arrival and Service Patterns The Poisson distribution often provides a reasonably good description of customer arrivals per unit of time (e. g. , per hour). § A discrete probability distribution § probability of a given number of events occurring in a fixed interval of time § events occur with a known average rate and independently. § The expected value and variance of (random) variable is equal to λ P(r) = (e-λ λr ) / r! where r = arrivals/time unit and λ = mean arrivals/time unit f(t) = μe – μt where t = service time and μ = mean service time m = the average number of customers who can be served per time period. Mean service time = 1/m 18 -12

System Characteristics: Queue Discipline § § First-come, first-served Priority Preferred (loyalty programs/fee-based) Reservation (appointment) 18 -13

Waiting line Models § Patient § Customers enter the waiting line and remain until served § Reneging § Waiting customers grow impatient and leave the line § Jockeying § Customers may switch to another line § Balking § Upon arriving, decide the line is too long and decide not to enter the line 18 -14

Waiting Line Performance § Waiting line performance relates to potential customer dissatisfaction and cost: § The average number of customers, either in line or in the system § The average time the customer waits, either in line or in the system § System utilization § The implied cost of a given level of capacity and its related waiting line § The probability that an arrival would have to wait 18 -15

Waiting Time vs. Utilization System utilization reflects the extent to which servers are busy 100 % Average number on time waiting in line utiliz not ation is a goa realist l ic 0 System Utilization 100% 18 -16

Queuing Models: Infinite-Source 1. Single channel, exponential service time 2. Single channel, constant service time 3. Multiple channel, exponential service time 4. Multiple priority service, exponential service time All of the above assume Poisson arrival rate, and average arrival and service rates are steady (i. e. , steady state) 18 -17

Basic Relationships of Infinite Source Queuing Model § 18 -18

Infinite Source 18 -19

LEGENDS § M/M/1 – Poisson Arrival Rate (the first M), Poisson (or exponential) service rate (the second M), and 1 server (the 1) § M/D/1 – Poisson Arrival Rate (the M), deterministic (constant) service rate (the D), and 1 server (the 1) § M/M/S - Poisson Arrival Rate (the first M), Poisson (or exponential) service rate (the second M), and multiple servers (the S) 18 -20

Infinite source Lq is a key value, and generally one of the first values we should calculate Equation will be different for M/M/1, M/D/1 models, M/M/S models Little’s Law applied in Waiting Line: Number of People Waiting in Line = (Average Customer Arrival Rate) x (Average Time in the System) 18 -21

Example: Infinite Source Customers arrive at a bakery at an average rate of 18 per hour on weekday mornings. The arrival distribution can be described by a Poisson distribution with a mean of 18. Each clerk can serve a customer in an average of three minutes; this time can be described by an exponential distribution with a mean of 3. 0 minutes. A. What are the arrival and service rates? B. Compute the average number of customers being served at any time. C. Suppose it has been determined that the average number of customers waiting in line is 8. 1. Compute the average number of customers in the system (i. e. , waiting in line or being served), the average time customers wait in line, and the average time in the system. D. Determine the system utilization for M = 1, 2, and 3 servers. 18 -22

Example: Infinite Source § λ=18 customers per hour § We need to change the service time to comparable hourly rate. 60 minutes per hour, and 3 minutes per customer means 20 customers per hour. μ = 20 customers/per hour 18 -23

Different models within Infinite Source Single Server, Exponential Service Time, M/M/1 The queue discipline is first-come, first-served, and it is assumed that the customer arrival rate can be approximated by a Poisson distribution and service time by a negative exponential distribution. There is no limit on length of queue 18 -24

M/M/1 Example An airline is planning to open a satellite ticket desk in a new shopping plaza, staffed by one ticket agent. It is estimated that requests for tickets and information will average 15 per hour, and requests will have a Poisson distribution. Service time is assumed to be exponentially distributed. Previous experience with similar satellite operations suggests that mean service time should average about three minutes per request. Determine each of the following: § § § a. System utilization. b. Percentage of time the server (agent) will be idle. c. The expected number of customers waiting to be served. d. The average time customers will spend in the system. e. The probability of zero customers in the system and the probability of four customers in the system. 18 -25

M/M/1 Example 18 -26

Different models within Infinite Source Single Server, Constant Service Time, M/D/1 Waiting lines are a consequence of random, highly variable arrival and service rates. If a system can reduce or eliminate the variability of either or both, it can shorten waiting lines noticeably. A case in point is a system with constant service time. The effect of a constant service time is to cut in half the average number of customers waiting in line The average time customers spend waiting in line is also cut in half. Similar improvements can be realized by smoothing arrival times (e. g. , by use of appointments). 18 -27

M/D/1 Example Wanda's Car Wash & Dry is an automatic, fiveminute operation with a single bay. On a typical Saturday morning, cars arrive at a mean rate of eight per hour, with arrivals tending to follow a Poisson distribution. Find § a. The average number of cars in line. § b. The average time cars spend in line and service 18 -28

M/D/1 Example 18 -29

Different models within Infinite Source Multiple Servers, M/M/S § Two or more servers are working independently. Use of the model involves the following assumptions: § 1. A Poisson arrival rate and exponential service time. § 2. Servers all work at the same average rate. § 3. Customers form a single waiting line (in order to maintain first-come, first-served processing). § The multiple-server formulas are more complex than the single-server formulas, especially the formulas for Lq and P 0. . 18 -30

M/M/S Equations 18 -31

Queuing Model: Finite Source § Calling population is limited § One person may be responsible for handling breakdowns on 15 machines § There may be more than one server or channel § Arrival rates are required to be Poisson and service times exponential. § arrival rate of customers here is affected by the length of the waiting line § the arrival rate decreases as the length of the line increases 18 -32

Finite Source Queuing Formulas and Notations 18 -33

Constraint Management Managers may be able to reduce waiting times by actively managing one or more system constraints. § Use temporary workers § Shift demand § Standardize the service § Look for a bottleneck 18 -34

Psychology of waiting § § § § Occupied time feels shorter than unoccupied time People want to get started Anxiety makes waits seem longer Uncertain waits are longer than known, finite waits Unexplained waits are longer than explained waits Unfair waits are longer than equitable waits The more valuable the service, the longer the customer will wait § Solo waits feel longer than group waits Source: David H. Maister, “The Psychology of Waiting Lines, ” davidmaister. com, blog, September 8, 2008 18 -35

Two key items to remember § Waiting lines forms because arrival rates and services rates are different. In other words, there is mismatch between supply and demand, or waiting lines happen due to the variability in arrival and service rates. § Increase in system utilization results in an exponential increase in customer waiting time. 18 -36

Disney’s approach to managing waiting lines § Provide distractions § Provide alternatives for those willing to pay a premium § Keep customers informed § Exceed expectations § Comfortable waiting environment and other distractions § A form of reservations (Fast Pass) 18 -37

Operations Strategy § Carefully assess the costs and benefits of various alternatives for capacity of service systems. § Increase the processing rate vs. number of servers. § New processing equipment/methods § Standardization (reduce variability in processing) § Shift some arrivals to “off-times” by using reservations systems § “early-bird” specials § senior discounts 18 -38